On Mixing Inequalities: Rank, Closure, and Cutting-Plane Proofs

We study the mixing inequalities which were introduced by Günlük and Pochet (2001). We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a type I mixing inequality and at most n− 1 if it is a type II mixing inequality. We also show that these bounds are tight for n = 2. Given a mixed-integer set PI = P ∩ Z(I) where P is a polyhedron and Z(I) = {x ∈ Rn : xi ∈ Z ∀i ∈ I}, we define mixing inequalities for PI . We show that the elementary mixing closure of P with respect to I can be described using a bounded number of mixing inequalities, each of which has a bounded number of terms. This implies that the elementary mixing closure of P is a polyhedron. Finally, we show that any mixing inequality can be derived via a polynomial length MIR cutting plane proof. Combined with results of Dash (2006) and Pudlák (1997), this implies that there are valid inequalities for a certain mixed-integer set that cannot be obtained via a polynomial-size mixing cutting-plane proof.